Question bank AMC AUSTRALIA 2013

$2013+2014+2015$ equals

In the diagram below, what is the value, in degrees, of angle x?

If every digit of a whole number is either a 3 or a 5, the number will always be

The average of two numbers is twice the smaller number. The larger number is 12. What is the smaller number?

The length of the base of a triangle is 3 times its perpendicular height and the area of the triangle is $24\text{ cm}^2$. The sum of its base length and its perpendicular height, in centimetres, is

A regular icosahedron is a solid shape with twenty faces, where each face is directly opposite another face. I label the faces from 1 to 20 so that, for all pairs of opposite faces, the two labels in any pair always add up to the same number. What number is on the face opposite the one labelled 8?

If $p=4b+26$ and b is a positive integer, then p could not be divisible by

My two dogs were running on the beach when I called them back. The faster dog was 100 m away and the slower dog was 70 m away. The faster dog runs twice as fast as the slower dog. How far away was the second dog when the first dog reached me?

The value of $x^2+\frac1{x^2}$ when $x=\frac23$ is closest to

A piece of paper in the shape of an equilateral triangle has one corner folded over, as shown.



What is the value of x?

Start with the number 1 and create the sequence

$$1,2,4,8,16,22,24,28,\ldots$$

where each number is the sum of the previous number and its final digit. How many numbers in the sequence are less than 1000?

A six-sided dice has the numbers 1, 2, 2, 3, 3 and 3 on its faces. Two such dice are rolled and a score is made by adding the numbers on the uppermost faces. The probability of rolling an odd score is

If $x^2=x+3$, then $x^3$ equals

The point T divides the side QR of the rectangle PQRS into two equal segments. The point U divides PQ such that $PU:UQ=1:2$. Point V divides SP such that $SV:VP=1:3$ and finally, point W divides RS such that $RW:WS=1:4$. Find the area of the quadrilateral TUVW if the area of PQRS equals 120.

Three line segments of lengths 1, a and 2a are the sides of a triangle. Which of the following defines all possible values of a?

The shaded segment in the circle below, centre O, has an area of $1\text{ cm}^2$. The radius of the circle, in centimetres, is

Dan and Jane each have a measuring tape of length 1 m. Dan’s tape got stuck in a door and was extended by 4 cm. Jane left her tape in a pocket and it shrank by 5 cm after washing. However, the centimetre marks on both tapes remained evenly distributed.

Measuring the schoolyard, Dan noted the length as 23.75 m. What length will Jane get measuring the same schoolyard with her tape?

In the regular hexagon pictured, the midpoints of the sides are joined to form the shaded regular hexagon. What fraction of the larger hexagon is shaded?

A circular wheel of radius r rolls, without slipping, through half a revolution. The point X is on the horizontal diameter at the start.



The distance between the starting and finishing position of the point X is

The sport of bingbong involves two players. Each match consists of a number of rounds and each round consists of a number of points. The first player to win four points in a round wins the round. The first player to win six rounds in a match wins the match.

Suppose that after a match of bingbong, the winner has won W points while the loser has won L points. What is the largest possible value of L − W?